\newproblem{lay:5_2_1}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.2.1}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Find the characteristic equation and the real eigenvalues of the matrix $A=\begin{pmatrix}2 & 7 \\ 7 & 2\end{pmatrix}$.
}{
   % Solution
	The characteristic equation is $|A-\lambda I|=0$. In this particular case
	\begin{center}
		$\left|\begin{array}{cc}2-\lambda & 7 \\ 7 & 2-\lambda\end{array}\right|=0$ \\
		$(2-\lambda)^2-49=0$\\
		$4+\lambda^2-4\lambda-49=0$\\
		$\lambda^2-4\lambda-45=0$ \\
		$\lambda=\frac{4\pm\sqrt{16+4\cdot 45}}{2}=\frac{4\pm 14}{2}=\left\{\begin{array}{r}9\\-5\end{array} \right.$
	\end{center}
	The two real eigenvalues are $\lambda=9$ and $\lambda=-5$.
}
\useproblem{lay:5_2_1}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
